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1Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden,Germany. 2Max Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle,Germany. 3Institut für Festkörperphysik, Technische Universität Darmstadt, 64289Darmstadt, Germany.*Corresponding author. E-mail: nayak@cpfs.mpg.de (A.K.N.); stuart.parkin@mpi-halle.mpg.de(S.S.P.P.)

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Large anomalous Hall effect driven by anonvanishing Berry curvature in the noncolinearantiferromagnet Mn3Ge

Ajaya K. Nayak,1,2* Julia Erika Fischer,1 Yan Sun,1 Binghai Yan,1 Julie Karel,1 Alexander C. Komarek,1

Chandra Shekhar,1 Nitesh Kumar,1 Walter Schnelle,1 Jürgen Kübler,3 Claudia Felser,1 Stuart S. P. Parkin2*

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It is well established that the anomalous Hall effect displayed by a ferromagnet scales with its magnetization.Therefore, an antiferromagnet that has no net magnetization should exhibit no anomalous Hall effect. We showthat the noncolinear triangular antiferromagnet Mn3Ge exhibits a large anomalous Hall effect comparable to that offerromagnetic metals; the magnitude of the anomalous conductivity is ~500 (ohm·cm)−1 at 2 K and ~50 (ohm·cm)−1

at room temperature. The angular dependence of the anomalous Hall effect measurements confirms that thesmall residual in-planemagnetic moment has no role in the observed effect except to control the chirality of the spintriangular structure.Our theoretical calculations demonstrate that the large anomalousHall effect inMn3Georiginatesfrom a nonvanishing Berry curvature that arises from the chiral spin structure, and that also results in a large spin Halleffect of 1100 (ħ/e) (ohm·cm)−1, comparable to that of platinum. The present results pave the way toward therealization of room temperature antiferromagnetic spintronics and spin Hall effect–based data storage devices.

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INTRODUCTION

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In a normal conductor, the Lorentz force gives rise to the Hall effect,wherein the induced Hall voltage varies linearly with the applied mag-netic field. Moreover, in a ferromagnetic material, one can realize alarge spontaneous Hall effect—termed the anomalous Hall effect(AHE)—that scales with its magnetization (1, 2). Although the detailedorigin of the AHE has been puzzling for many years, recent theoreticaldevelopments have provided a deeper understanding of the intrinsiccontribution of the AHE in ferromagnets in terms of the Berry-phasecurvature (1, 3–6). The Berry phase is a very useful tool for understand-ing the evolution of an electron’s wave function in the presence of a driv-ing perturbation, such as an electric field. The electron’s motion maydeviate from its usual path, wherein the electron follows a path along thedirection of the applied electric field, and may pick up a transverse compo-nent of momentum even in the absence of an external magnetic field,leading to a Hall effect. This process can be understood by calculating theBerry curvature, which is an intrinsic property of the occupied electronicstates in a given crystal with a certain symmetry, and only takes a nonzerovalue in systems where time-reversal symmetry is broken (for example,by the application of a magnetic field) or where magnetic moments arepresent. From symmetry analysis, the Berry curvature vanishes for con-ventional antiferromagnets with colinear moments, thereby resulting inzero AHE. However, a nonvanishing Berry phase in antiferromagnetswith a noncolinear spin arrangement was recently predicted to lead toa large AHE (7–9). In particular, Chen et al. (7) have demonstrated the-oretically that an anomalous Hall conductivity (AHC) larger than 200(ohm·cm)−1 can be obtained in the noncolinear antiferromagnet Mn3Ir.

The Heusler compound Mn3Ge is known to crystallize in bothtetragonal and hexagonal structures, depending on the annealingtemperature (10, 11). Hexagonal Mn3Ge exhibits a triangular anti-

ferromagnetic structure with an ordering temperature of ~365 to400 K (11–14). As shown in Fig. 1A, the hexagonal unit cell of Mn3Geconsists of two layers of Mn triangles stacked along the c axis. In eachlayer, the Mn atoms form a Kagome lattice, with Ge sitting at thecenter of a hexagon. Neutron diffraction studies have revealed a non-colinear triangular antiferromagnetic spin arrangement of the Mnspins in which the neighboring moments are aligned in-plane at anangle of 120° (13, 14). These same studies have reported that the spintriangle is free to rotate in an external magnetic field (13, 14). This non-colinear structure arises as a result of the geometrical frustration of theMn moments that are arranged in a triangular configuration in the a-bplane. We have performed ab initio calculations of the magnetic groundstate of Mn3Ge. The lowest energy spin configuration is shown in Fig. 1Band is consistent with previous calculations (15, 16). In addition to the120° ordered antiferromagnetic structure, a very small net in-plane mo-ment exists as a result of a small tilting of the Mn moments (13).

RESULTS AND DISCUSSION

We have calculated the Berry curvature and AHC in Mn3Ge using thelinear-response Kubo formulism (1) based on our calculated magneticstructure (Fig. 1B). We find that the AHC exhibits nearly zero valuesfor both xy (sz

xy) and yz (sxyz) components and only large values for the

xz component syxz = 330 (ohm·cm)−1. Here, skij corresponds to the

AHC when current flows along the j direction, and thereby resultsin a Hall voltage along the i direction: skij can be treated as a pseudo-vector pointing along the k direction that is perpendicular to both the idirection and the j direction. The symmetry of the antiferromagneticstructure of Mn3Ge determines which components of the AHC tensormust be zero. As shown in Fig. 1, the two magnetic layers that com-pose the primitive unit cell of the antiferromagnetic structure ofMn3Ge can be transformed into each other by a mirror reflection,with respect to the xz plane (as indicated in Fig. 1), plus a translationby c/2 along the c axis. Because of the mirror symmetry, the components

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of the skij vanish if they align parallel to the mirror plane. This is the casefor szxy and sxyz , which are expected to be exactly zero. However, theaforementioned residual net in-plane moment acts as a perturbationof the mirror symmetry. As a consequence, szxy and sxyz can pick upnonzero but tiny values. By contrast, the component of skij perpendic-ular to the mirror plane (namely, sy

xz ) is nonzero. We show thecalculated values of sy

xz in the xz plane in Fig. 1D. The distributionis highly nonuniform and is characterized by four well-defined peaks(that is, “hot spots”). These peaks correspond to regions in k spacewhere there is a small energy gap between the occupied bands andthe empty bands: this small gap is a consequence of the spin-orbitcoupling within the Mn 3d band.

Moreover, a large AHC [~900 (ohm·cm)−1] has previously beenshown theoretically by considering a slightly different spin config-uration (8). In the latter case, a nonvanishing component of the AHCcan be obtained in the yz plane. The reason for this difference is straight-forward; in the spin configuration shown in Fig. 1B, if the spins parallelto y tilt slightly in the x direction, a nonzero Berry curvature in the yzplane will result. In all of these configurations, the AHE is zero in thexy plane if one strictly considers only an in-plane spin structure. Wenote that other noncolinear spin structures, including a 120° spinstructure on a triangular lattice, can also give rise to an AHE (9).

Like the AHE, the intrinsic spin Hall effect also originates from theband structure. The strong entanglement between occupied and

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unoccupied states near the Fermi level should give rise to a largecontribution to both anomalous and spin Hall conductivities. Mo-tivated by our large AHC, we have also theoretically analyzed thespin Hall conductivity (SHC), and we predict that a large SHC of1100 (ħ/e) (ohm·cm)−1 can be obtained in Mn3Ge. The theoreticalSHC in Mn3Ge is comparable to that of Pt (17) and is larger thanthe SHC recently reported in Mn3Ir, which also exhibits a triangu-lar antiferromagnetic structure (18). In the present work, we exper-imentally show evidence for an AHC in a hexagonal single crystalof Mn3Ge that agrees well with our theoretical predictions of AHC.

To probe the AHE in Mn3Ge, we have carried out detailed Halleffect measurements as a function of magnetic field over temperaturesvarying from 2 to 400 K. Figure 2A shows the Hall resistivity rH ver-sus the magnetic fieldHmeasured for the current (I) along [0001] and(H) parallel to [01-10]. rH saturates in modest magnetic fields, attain-ing a large saturation value of 5.1 mohm·cm at 2 K and ~1.8 mohm·cmat 300 K. The Hall conductivity for the same I and H configurations(which we label configuration I) (sxz), calculated from sH = −rH/r

2 (wherer is the normal resistivity), exhibits a large value of ~500 (ohm·cm)−1

at 2 K and a modest value of ~50 (ohm·cm)−1 at 300 K (Fig. 2B). Be-cause both rH and sH saturate in magnetic fields, these clearly reflectan AHE that is large in magnitude. To study whether the experi-mental AHE has an anisotropy, as predicted by theory, we measuredrH with I along [01-10] and with H parallel to [2-1-10] (syz). In this

Fig. 1. Crystal structure, magnetic structure, and Berry curvature of Mn3Ge. (A) Crystal structure showing the two layers of Mn-Ge atoms stackedalong the c(z) axis. Red and blue spheres represent atoms lying in the z = 0 and z = c/2 planes, respectively. Large and small spheres represent Mnand Ge atoms, respectively. In each layer, the Mn atoms form a Kagome-type lattice. (B) Calculated 120° antiferromagnetic configuration in thez = 0 and z = c/2 planes, respectively. Only the Mn atoms are shown, with their moments represented by arrows. The green dashed line in-dicates the mirror plane (that is, the xz plane). Corresponding mirror reflection plus a translation by c/2 along the z axis transforms the systemback into itself, by exchanging two magnetic planes. (C) First Brillouin zone and momentum-dependent AHC. The mirror plane (that is, kx − kzplane) is highlighted in green. The component of the AHC tensor (skij) is shown in the ki − kj plane, which corresponds to the Berry curvature ofall occupied electronic states integrated over the kk direction. Here, only the syxz component exhibits large values. (D) Illustration of therotation of the triangular spin structure with an in-plane field. There is a small in-plane net magnetic moment that we calculate to be alongthe direction indicated. When the field (H) rotates by 180°, the chiral spin structure can be reversed, resulting in the sign change in the AHC.

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configuration II, rH is slightly smaller than configuration I (~4.8 mohm·cmat 2 K and 1.6 mohm·cm at 300 K) (Fig. 2C). Although sH (syz) is re-duced at 2 K in configuration II compared to configuration I [to about150 (ohm·cm)−1], a similar value of sH = 50 (ohm·cm)−1 at 300 K isfound for both configuration I and configuration II (Fig. 2D). In bothcases, rH is negative (positive) for positive (negative) fields. Finally,in a third configuration (configuration III), we apply I along [2-1-10]andH parallel to [0001]. In this configuration, we observe small valuesof both rH and sH (sxy) at all temperatures (Fig. 2, E and F); more-over, the sign of AHE is opposite to that for configurations I and II.

To confirm that the observation of a large AHE in the present sys-tem originates from the noncolinear antiferromagnetic spin structure,we have performed magnetization measurements with the field par-allel to different crystallographic directions. The temperature depen-dence of magnetization M (T) indicates a TC of 380 K for the presentMn3Ge single crystal (see the Supplementary Materials), only slightlylower than that reported in the literature. TheM (H) loop measured withthe field parallel to [2-1-10] displays a tiny spontaneous magnetizationof 0.005 mB/Mn at 2 K (Fig. 3). The zero field moment is slightly larger(0.006 mB/Mn at 2 K) when the field is applied parallel to [01-10]. Anextremely soft magnetic behavior is found in both cases, with coercivefields (HC) of less than 20 Oe. These magnetization values agreewell with previous results that show an in-plane magnetic moment

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of ~0.007 mB/Mn (12). Although the Mn moments are expected to lieonly in the a-b plane, we observe a moment that is an order of mag-nitude less than 0.0007 mB/Mn at 2 K for a field parallel to [0001]. Thisindicates that there could be a very small tilting of the Mn spinstoward the c axis. The M (H) loops measured at 300 K for differentfield orientations are similar to the 2 K data, except that the high-field magnetization strongly decreases for H parallel to [0001] (inset toFig. 3).

From our magnetization data, it is clear that the present Mn3Gesample has a nearly zero net magnetic moment. Hence, we concludethat the extremely large AHE depicted in Fig. 2 must be related to thenoncolinear antiferromagnetic spin structure, as predicted by theory.The observation of a large AHC in the xz and yz planes, and of a verysmall effect in the xy plane, is also consistent with our theoretical calcu-lations. Although our calculations show a nonzero AHC only in the xzplane, this is true for the particular spin configuration considered (seeFig. 1B). By symmetry, the x and y directions must be symmetry-relatedso that there will be equal contributions to the measured AHE for thevarious symmetry-related spin configurations (by rotation by ±120° inFig. 1B) or the spin configurations can be rotated by the applied field asa result of the small in-plane magnetic moment, as illustrated in Fig. 1D.

It is interesting to compare the magnitude of the present AHE withthat exhibited by ferromagnetic Heusler compounds because the hex-agonal Mn3Ge structure evolves from the cubic Heusler structure bydistortion of the lattice along the (111) direction. For example, theferromagnetic Heusler compounds Co2MnSi and Co2MnGe, whichexhibit a net magnetic moment of ~5 mB/Mn (more than two ordersof magnitude higher by comparison to the present h-Mn3Ge), dis-play a maximum rH of ~4 mohm·cm (19). The same study showedthat rH decreases by two orders of magnitude when the magnetizationis reduced to nearly half in Cu2MnAl (19). A similar scaling of rH withmagnetization has also been demonstrated by Zeng et al. (2) for fer-romagnetic Mn5Ge3. Moreover, it has also been well established thatthe Hall conductivity in many ferromagnetic materials scales with theirnormal conductivity (19–21). In the present case, the Hall conductivitymeasured with the current along different crystallographic directions

Fig. 2. AHE. (A to F) Hall resistivity (rH) (A, C, and E) and Hall conduc-tivity (sH) (B, D, and F) as a function of magnetic field (H), measured atfour representative temperatures between 2 and 300 K, for three differ-ent current and magnetic field configurations. The hexagon in the inset to(C) and (D) shows the two in-plane directions used for the current andmagnetic field in the present measurements.

Fig. 3. Magnetic properties. Field dependence of magnetization M (H)measured at 2 K, with the field parallel to the three orientations used inthe AHE measurements. The inset shows M (H) loops measured at 300 K.

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follows a behavior similar to that of the normal conductivity (see theSupplementary Materials). Similar data for the AHE in the noncolinearantiferromagnet Mn3Sn have been recently published by Nakatsuji et al.(22). However, Mn3Ge exhibits distinctions in its magnetism, as wellas its AHE, in comparison to Mn3Sn. In particular, the maximum AHCin the present system is nearly three times higher than that of Mn3Sn.Moreover, Mn3Ge does not show any secondary transition, such as thecluster glass phase observed at low temperatures in Mn3Sn (22).

According to theoretical predictions, the nonvanishing Berry cur-vature arising from the noncolinear triangular antiferromagneticstructure is responsible for the observed large AHE. In such a scenario,we would expect to observe an AHE without applying any magneticfield. However, the special spin arrangement in both h-Mn3Ge andh-Mn3Sn can lead to the formation of antiferromagnetic domains.Therefore, in zero magnetic field, the direction of the Hall voltage maybe different in each antiferromagnetic domain, resulting in a vanishingAHC. The presence of a weak in-plane ferromagnetic moment ensuresthat a small magnetic field in one direction within the a-b plane issufficient to eliminate the domains, and hence allows the establishmentof a large AHE. In particular, neutron diffraction measurements haveestablished that an applied magnetic field does not substantially af-fect the 120° triangular antiferromagnetic spin structure, but the mag-netic field does result in a rotation of the spin triangle opposite to thein-plane field component (14, 23). Therefore, switching the directionof an in-plane field from positive to negative results in the switching ofthe direction of the staggered moment in the triangular spin structure,and consequently causes a change in the sign of the anomalous Hallvoltage. The small variation of the anomalous Hall signal at high fieldslikely originates from the normal component of the Hall signal or fromthe magnetic susceptibility of the magnetic structure.

To establish that the presence of the weak in-plane ferromagnetismin Mn3Ge has virtually no role in the observed AHE, we have carriedout angular-dependent measurements of the AHE for various fieldand current configurations. In configuration I shown in Fig. 4A, thecurrent was applied along z and the voltage was measured in the xzplane, resulting in Vx. In this configuration, the initial field directionfor q = 0 is along y. The field was then rotated in the y-z plane. Underthese conditions, the AHC (syz) remains constant up to q = 90°, whereit suddenly changes sign. A similar sign change occurs again at q =270°. This behavior can be explained as follows: with increasing q, al-though the in-plane magnetic field decreases, a small component ofthe field lies parallel to the a-b plane up to q = 90°. This small in-planefield maintains the chirality of the triangular spin structure in a singledirection. At angles greater than 90°, the in-plane field changes its di-rection, resulting in a change in the chirality of the spin structure,which produces the sign change in the AHE. A similar phenomenonhappens at 270° (Fig. 4B). In configuration II, we apply the currentalong y with the voltage measured in the yz plane (Vz), and the fieldis rotated in the x-z plane. In this case, H is always perpendicular to I.The angular dependence of the AHC (syz) follows the same behavioras that of configuration I. In both cases, a large and constant AHC wasobtained when at least a small field parallel to the in-plane spin struc-ture was present. We now show that the AHC measured in the xyplane (sxy) always remains nearly zero even when there is a nonzerofield component parallel to the plane of the triangular spin structure.In configuration III, Vy was measured in the xy plane with I along x(Fig. 4C). The field was rotated in the z-y plane. In this configu-ration, a component of the magnetic field always remains parallel

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to the a-b plane, except for q = 0 and 180°. However, sxy remainsnearly zero regardless of the field direction. A small nonvanishingvalue of sxy may appear from a slight canting of the spins along thez (c) direction. The feasibility of a nonplanar structure in Mn3Ge hasalso been discussed in our previous theoretical work (8). From theseangle-dependent measurements, it can be readily concluded thatthe small residual in-plane ferromagnetic component, as well as the ap-plied magnetic field, plays no role in the observed large AHE in Mn3Ge.

In summary, we have experimentally demonstrated that the non-colinear antiferromagnet Mn3Ge exhibits a large AHE, even at roomtemperature. The origin of this exotic effect is discussed in terms of anonvanishing Berry curvature in momentum space. Antiferromagnetsare of significant interest for spintronic applications (24–26) because,in contrast to ferromagnets, they do not produce any undesirable di-pole fields. Therefore, the present antiferromagnetic material with alarge AHE, which is easily controlled by the current, can be exploitedfor spintronics, without the drawbacks of ferromagnets. Finally, ourtheoretical finding of a large spin Hall effect in Mn3Ge will motivate

Fig. 4. Angular dependence of AHE. Angular (q) dependence of theAHC measured for the three current and field configurations used in Fig. 2.A schematic diagram of the sample geometry is shown for each con-figuration. (A to C) The rotation was performed along (A) y-z for I alongz [0001], (B) x-z for I along y [01-10], and (C) y-z for I along x [2-1-10]. q =0 corresponds to H oriented along y, x, and z in (A), (B), and (C), respec-tively. We have used the same x, y, z coordinates as shown in the theoreticalspin arrangement in Fig. 1B.

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further studies to explore other phenomena derived from the Berry-phase curvature in antiferromagnets.

During the writing of the present manuscript, we became aware ofa similar study of a large AHE in Mn3Sn conducted by Nakatsuji et al.(22). We have compared the present findings for Mn3Ge with thosefor Mn3Sn in the main text.

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MATERIALS AND METHODS

For the single-crystal growth, stoichiometric quantities of pure ele-ments were weighed and premelted in an alumina crucible in an induc-tion furnace. The ingot was ground and put in an alumina crucible,which was sealed in a quartz ampule. The single crystal was grownusing the Bridgman-Stockbarger technique. The growth temperaturewas controlled at the bottom of the ampule. The material was heatedup to 980°C, remained there for 1 hour to ensure homogeneity of themelt, and was cooled down slowly to 740°C. The sample was annealedat 740°C for 15 hours and then quenched to room temperature in argongas. The sample was cut into five slices perpendicular to the growthdirection. The dimensions of each slice were approximately 6 mm indiameter and 2 mm thick. Single-crystal x-ray diffraction was carriedout on a Bruker D8 VENTURE x-ray diffractometer using Mo-K radia-tion and a bent graphite monochromator. The magnetization mea-surements were carried out on a vibrating sample magnetometer (MPMS3, Quantum Design). The transport measurements were performed withlow-frequency alternating current (ACT option, PPMS 9, Quantum De-sign). Our ab initio density functional theory calculations were performedusing the Vienna Ab-initio Simulation Package with projected augmentedwave potential (27). The exchange-correlation energy was considered atthe generalized gradient approximation level (28). Further details of thecalculations are given in the Supplementary Materials.

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SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/4/e1501870/DC1fig. S1. Temperature dependence of magnetization.fig. S2. Conductivity with temperature.

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Acknowledgments: We thank C. Geibel for helpful discussions and use of the Bridgman fur-nace. Funding: This work was financially supported by the Deutsche Forschungsgemeinschaft(project numbers TP 1.2-A and 2.3-A of Research Unit FOR 1464 ASPIMATT) and by the Euro-pean Research Council Advanced Grant (grant 291472) “Idea Heusler.” Author contributions:A.K.N. performed the transport and magnetic measurements with the help of C.S., N.K., andW.S. The theoretical calculations were carried out by Y.S., B.Y., and J. Kübler. J.E.F. synthesizedthe single crystal. J.E.F., A.C.K., and J. Karel performed the structural characterization. A.K.N.and S.S.P.P. wrote the manuscript, with substantial contributions from all authors. C.F. and S.S.P.P.inspired and jointly supervised the project. Competing interests: The authors declare that theyhave no competing interests. Data and materials availability: All data used to obtain theconclusions in this paper are presented in the paper and/or the Supplementary Materials. Otherdata may be requested from the authors. Please direct all inquiries to A.K.N. (nayak@cpfs.mpg.de)and S.S.P.P. (stuart.parkin@mpi-halle.mpg.de).

Submitted 20 December 2015Accepted 17 March 2016Published 15 April 201610.1126/sciadv.1501870

Citation: A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel, A. C. Komarek, C. Shekhar, N. Kumar,W. Schnelle, J. Kübler, C. Felser, S. S. P. Parkin, Large anomalous Hall effect driven by anonvanishing Berry curvature in the noncolinear antiferromagnet Mn3Ge. Sci. Adv. 2,e1501870 (2016).

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Ge3antiferromagnet MnLarge anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear

Kumar, Walter Schnelle, Jürgen Kübler, Claudia Felser and Stuart S. P. ParkinAjaya K. Nayak, Julia Erika Fischer, Yan Sun, Binghai Yan, Julie Karel, Alexander C. Komarek, Chandra Shekhar, Nitesh

DOI: 10.1126/sciadv.1501870 (4), e1501870.2Sci Adv 

ARTICLE TOOLS http://advances.sciencemag.org/content/2/4/e1501870

MATERIALSSUPPLEMENTARY http://advances.sciencemag.org/content/suppl/2016/04/11/2.4.e1501870.DC1

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